Healthcare is controlled by the government because it enforces licensure artificially limiting the number of doctors. Prior to the AMA's lobbying for such restrictions in the early 1900s, medical care was cheap and plentiful; people even had doctors go to their houses.
Education is also artificially influenced by the government because it gives out special loans and makes it illegal to declare bankruptcy on them, allowing colleges to keep raising prices because they know students will always be able to pay via student loans.
Those issues are still the result of the market responding to the government's actions/policies; remove the market (make both single payer) and the government will have much more incentive to fix the problems.
The market has a response to literally everything. You can't 'remove the market' from a given industry. Single Payer would still have to pay market prices.
Education is also artificially influenced by the government because it gives out special loans and makes it illegal to declare bankruptcy on them, allowing colleges to keep raising prices because they know students will always be able to pay via student loans.